Foundations of Artificial Intelligence
|
|
- Neil Fitzgerald
- 6 years ago
- Views:
Transcription
1 Foundations of Artificial Intelligence 8. Satisfiability and Model Construction Davis-Putnam-Logemann-Loveland Procedure, Phase Transitions, GSAT Joschka Boedecker and Wolfram Burgard and Bernhard Nebel Albert-Ludwigs-Universität Freiburg May 26, 2017
2 Contents 1 Motivation 2 Davis-Putnam-Logemann-Loveland (DPLL) Procedure 3 Average complexity of the satisfiability problem 4 GSAT: Greedy SAT Procedure (University of Freiburg) Foundations of AI May 26, / 24
3 Motivation Propositional Logic typical algorithmic questions: Logical deduction Given: A logical theory (set of propositions) Question: Does a proposition logically follow from this theory? Reduction to unsatisfiability, which is conp-complete (complementary to NP problems) Satisfiability of a formula (SAT) Given: A logical theory Wanted: Model of the theory Example: Configurations that fulfill the constraints given in the theory Can be easier because it is enough to find one model (University of Freiburg) Foundations of AI May 26, / 24
4 The Satisfiability Problem (SAT) Given: Propositional formula ϕ in CNF Wanted: Model of ϕ. or proof, that no such model exists. (University of Freiburg) Foundations of AI May 26, / 24
5 SAT and CSP SAT can be formulated as a Constraint-Satisfaction-Problem ( search): (University of Freiburg) Foundations of AI May 26, / 24
6 SAT and CSP SAT can be formulated as a Constraint-Satisfaction-Problem ( search): CSP-Variables = Symbols of the alphabet Domain of values = {T, F } Donstraints given by clauses (University of Freiburg) Foundations of AI May 26, / 24
7 The DPLL algorithm The DPLL algorithm (Davis, Putnam, Logemann, Loveland, 1962) corresponds to backtracking with inference in CPSs: recursive Call DPLL (, l) with : set of clauses and l: variable assignment result is a satisfying assignment that extends l or unsatisfiable if no such assignment exists. first call by DPLL(, ) Inference in DPLL: simplify: if variable v is assigned a value d, then all clauses containing v are simplified immediately (corresponds to forward checking) variables in unit clauses (= clauses with only one variable) are immediately assigned (corresponds to minimum remaining values ordering in CSPs) (University of Freiburg) Foundations of AI May 26, / 24
8 The DPLL Procedure DPLL Function Given a set of clauses defined over a set of variables Σ, return satisfiable if is satisfiable. Otherwise return unsatisfiable. 1. If = return satisfiable 2. If return unsatisfiable 3. Unit-propagation Rule: If contains a unit-clause C, assign a truth-value to the variable in C that satisfies C, simplify to and return DPLL( ). 4. Splitting Rule: Select from Σ a variable v which has not been assigned a truth-value. Assign one truth value t to it, simplify to and call DPLL( ) a. If the call returns satisfiable, then return satisfiable. b. Otherwise assign the other truth-value to v in, simplify to and return DPLL( ). (University of Freiburg) Foundations of AI May 26, / 24
9 Example (1) = {{a, b, c}, { a, b}, {c}, {a, b}} (University of Freiburg) Foundations of AI May 26, / 24
10 Example (1) = {{a, b, c}, { a, b}, {c}, {a, b}} 1. Unit-propagation rule: c T (University of Freiburg) Foundations of AI May 26, / 24
11 Example (1) = {{a, b, c}, { a, b}, {c}, {a, b}} 1. Unit-propagation rule: c T {{a, b}, { a, b}, {a, b}} (University of Freiburg) Foundations of AI May 26, / 24
12 Example (1) = {{a, b, c}, { a, b}, {c}, {a, b}} 1. Unit-propagation rule: c T {{a, b}, { a, b}, {a, b}} 2. Splitting rule: (University of Freiburg) Foundations of AI May 26, / 24
13 Example (1) = {{a, b, c}, { a, b}, {c}, {a, b}} a. a F {{b}, { b}} 1. Unit-propagation rule: c T {{a, b}, { a, b}, {a, b}} 2. Splitting rule: (University of Freiburg) Foundations of AI May 26, / 24
14 Example (1) = {{a, b, c}, { a, b}, {c}, {a, b}} a. a F {{b}, { b}} 1. Unit-propagation rule: c T {{a, b}, { a, b}, {a, b}} 2. Splitting rule: a. Unit-propagation rule: b T { } (University of Freiburg) Foundations of AI May 26, / 24
15 Example (1) = {{a, b, c}, { a, b}, {c}, {a, b}} a. a F {{b}, { b}} 1. Unit-propagation rule: c T {{a, b}, { a, b}, {a, b}} 2. Splitting rule: a. Unit-propagation rule: b T { } 2b. a T {{ b}} (University of Freiburg) Foundations of AI May 26, / 24
16 Example (1) = {{a, b, c}, { a, b}, {c}, {a, b}} a. a F {{b}, { b}} 1. Unit-propagation rule: c T {{a, b}, { a, b}, {a, b}} 2. Splitting rule: a. Unit-propagation rule: b T { } 2b. a T {{ b}} 3b. Unit-propagation rule: b F {} (University of Freiburg) Foundations of AI May 26, / 24
17 Example (1) = {{a, b, c}, { a, b}, {c}, {a, b}} a. a F {{b}, { b}} 1. Unit-propagation rule: c T {{a, b}, { a, b}, {a, b}} 2. Splitting rule: a. Unit-propagation rule: b T { } 2b. a T {{ b}} 3b. Unit-propagation rule: b F {} (University of Freiburg) Foundations of AI May 26, / 24
18 Example (2) = {{a, b, c, d}, {b, d}, {c, d}, {d}} (University of Freiburg) Foundations of AI May 26, / 24
19 Example (2) = {{a, b, c, d}, {b, d}, {c, d}, {d}} 1. Unit-propagation rule: d T (University of Freiburg) Foundations of AI May 26, / 24
20 Example (2) = {{a, b, c, d}, {b, d}, {c, d}, {d}} 1. Unit-propagation rule: d T {{a, b, c}, {b}, {c}} (University of Freiburg) Foundations of AI May 26, / 24
21 Example (2) = {{a, b, c, d}, {b, d}, {c, d}, {d}} 1. Unit-propagation rule: d T {{a, b, c}, {b}, {c}} 2. Unit-propagation rule: b T {{a, c}, {c}} (University of Freiburg) Foundations of AI May 26, / 24
22 Example (2) = {{a, b, c, d}, {b, d}, {c, d}, {d}} 1. Unit-propagation rule: d T {{a, b, c}, {b}, {c}} 2. Unit-propagation rule: b T {{a, c}, {c}} 3. Unit-propagation rule: c T {{a}} (University of Freiburg) Foundations of AI May 26, / 24
23 Example (2) = {{a, b, c, d}, {b, d}, {c, d}, {d}} 1. Unit-propagation rule: d T {{a, b, c}, {b}, {c}} 2. Unit-propagation rule: b T {{a, c}, {c}} 3. Unit-propagation rule: c T {{a}} 4. Unit-propagation rule: a T {} (University of Freiburg) Foundations of AI May 26, / 24
24 Example (2) = {{a, b, c, d}, {b, d}, {c, d}, {d}} 1. Unit-propagation rule: d T {{a, b, c}, {b}, {c}} 2. Unit-propagation rule: b T {{a, c}, {c}} 3. Unit-propagation rule: c T {{a}} 4. Unit-propagation rule: a T {} (University of Freiburg) Foundations of AI May 26, / 24
25 Properties of DPLL DPLL is complete, correct, and guaranteed to terminate. DPLL constructs a model, if one exists. In general, DPLL requires exponential time (splitting rule!) Heuristics are needed to determine which variable should be instantiated next and which value should be used. DPLL is polynomial on Horn clauses, i.e., clauses with at most one positive literal A 1,... A n B (see next slides). In all SAT competitions so far, DPLL-based procedures have shown the best performance. (University of Freiburg) Foundations of AI May 26, / 24
26 DPLL on Horn Clauses (0) Horn Clauses constitute an important special case, since they require only polynomial runtime of DPLL. Definition: A Horn clause is a clause with maximally one positive literal E.g., A 1... A n B or A 1... A n (n = 0 is permitted). Equivalent representation: A 1... A n B i A i B Basis of logic programming (e.g., PROLOG) (University of Freiburg) Foundations of AI May 26, / 24
27 DPLL on Horn Clauses (1) Note: 1. The simplifications in DPLL on Horn clauses always generate Horn clauses 2. If the first sequence of applications of the unit propagation rule in DPLL does not lead to termination, a set of Horn clauses without unit clauses is generated 3. A set of Horn clauses without unit clauses and without the empty clause is satisfiable, since All clauses have at least one negative literal (since all non-unit clauses have at least two literals, where at most one can be positive (Def. Horn)) Assigning false to all variables satisfies formula (University of Freiburg) Foundations of AI May 26, / 24
28 DPLL on Horn Clauses (2) 4. It follows from 3.: a. every time the splitting rule is applied, the current formula is satisfiable b. every time, when the wrong decision (= assignment in the splitting rule) is made, this will be immediately detected (e.g., only through unit propagation steps and the derivation of the empty clause). 4. Therefore, the search trees for n variables can only contain a maximum of n nodes, in which the splitting rule is applied (and the tree branches). 4. Therefore, the size of the search tree is only polynomial in n and therefore the running time is also polynomial. (University of Freiburg) Foundations of AI May 26, / 24
29 How Good is DPLL in the Average Case? We know that SAT is NP-complete, i.e., in the worst case, it takes exponential time. This is clearly also true for the DPLL-procedure. Couldn t we do better in the average case? For CNF-formulae, in which the probability for a positive appearance, negative appearance and non-appearance in a clause is 1/3, DPLL needs on average quadratic time (Goldberg 79)! The probability that these formulae are satisfiable is, however, very high. (University of Freiburg) Foundations of AI May 26, / 24
30 Phase Transitions... Conversely, we can, of course, try to identify hard to solve problem instances. Cheeseman et al. (IJCAI-91) came up with the following plausible conjecture: All NP-complete problems have at least one order parameter and the hard to solve problems are around a critical value of this order parameter. This critical value (a phase transition) separates one region from another, such as over-constrained and under-constrained regions of the problem space. Confirmation for graph coloring and Hamilton path..., later also for other NP-complete problems. (University of Freiburg) Foundations of AI May 26, / 24
31 Phase Transitions with 3-SAT Constant clause length model (Mitchell et al., AAAI-92): Clause length k is given. Choose variables for every clause k and use the complement with probability 0.5 for each variable. Phase transition for 3-SAT with a clause/variable ratio of approx. 4.3: (University of Freiburg) Foundations of AI May 26, / 24
32 Empirical Difficulty The Davis-Putnam (DPLL) Procedure shows extreme runtime peaks at the phase transition: Note: Hard instances can exist even in the regions of the more easily satisfiable/unsatisfiable instances! (University of Freiburg) Foundations of AI May 26, / 24
33 Notes on the Phase Transition When the probability of a solution is close to 1 (under-constrained), there are many solutions, and the first search path of a backtracking search is usually successful. If the probability of a solution is close to 0 (over-constrained), this fact can usually be determined early in the search. In the phase transition stage, there are many near successes ( close, but no cigar ) (limited) possibility of predicting the difficulty of finding a solution based on the parameters (search intensive) benchmark problems are located in the phase region (but they have a special structure) (University of Freiburg) Foundations of AI May 26, / 24
34 Local Search Methods for Solving Logical Problems In many cases, we are interested in finding a satisfying assignment of variables (example CSP), and we can sacrifice completeness if we can solve much large instances this way. Standard process for optimization problems: Local Search Based on a (random) configuration Through local modifications, we hope to produce better configurations Main problem: local maxima (University of Freiburg) Foundations of AI May 26, / 24
35 Dealing with Local Maxima As a measure of the value of a configuration in a logical problem, we could use the number of satisfied constraints/clauses. But local search seems inappropriate, considering we want to find a global maximum (all constraints/clauses satisfied). By restarting and/or injecting noise, we can often escape local maxima. Actually: Local search performs very well for finding satisfying assignments of CNF formulae (even without injecting noise). (University of Freiburg) Foundations of AI May 26, / 24
36 GSAT Procedure GSAT INPUT: a set of clauses α, Max-Flips, and Max-Tries OUTPUT: a satisfying truth assignment of α, if found begin for i := 1 to Max-Tries T := a randomly-generated truth assignment for j := 1 to Max-Flips if T satisfies α then return T v := a propositional variable such that a change in its truth assignment gives the largest increase in the number of clauses of α that are satisfied by T T := T with the truth assignment of v reversed end for end for return no satisfying assignment found end (University of Freiburg) Foundations of AI May 26, / 24
37 The Search Behavior of GSAT In contrast to normal local search methods, we must also allow sideways movements! Most time is spent searching on plateaus. (University of Freiburg) Foundations of AI May 26, / 24
38 State of the Art SAT competitions since beginning of the 90s Current SAT competitions ( In 2010: Largest industrial instances: > 1,000,000 literals Complete solvers are as good as randomized ones on handcrafted and industrial problem (University of Freiburg) Foundations of AI May 26, / 24
39 Concluding Remarks DPLL-based SAT solvers prevail: Very efficient implementation techniques Good branching heuristics Clause learning Incomplete randomized SAT-solvers are good (in particular on random instances) but there is no dramatic increase in size of what they can solve parameters are difficult to adjust (University of Freiburg) Foundations of AI May 26, / 24
Foundations of Artificial Intelligence
Foundations of Artificial Intelligence 32. Propositional Logic: Local Search and Outlook Martin Wehrle Universität Basel April 29, 2016 Propositional Logic: Overview Chapter overview: propositional logic
More informationFoundations of Artificial Intelligence
Foundations of Artificial Intelligence 31. Propositional Logic: DPLL Algorithm Malte Helmert and Gabriele Röger University of Basel April 24, 2017 Propositional Logic: Overview Chapter overview: propositional
More informationArtificial Intelligence
Torralba and Wahlster Artificial Intelligence Chapter 11: Propositional Reasoning, Part II 1/69 Artificial Intelligence 11. Propositional Reasoning, Part II: SAT Solvers How to Efficiently Think About
More informationCOMP219: Artificial Intelligence. Lecture 20: Propositional Reasoning
COMP219: Artificial Intelligence Lecture 20: Propositional Reasoning 1 Overview Last time Logic for KR in general; Propositional Logic; Natural Deduction Today Entailment, satisfiability and validity Normal
More informationChapter 7 R&N ICS 271 Fall 2017 Kalev Kask
Set 6: Knowledge Representation: The Propositional Calculus Chapter 7 R&N ICS 271 Fall 2017 Kalev Kask Outline Representing knowledge using logic Agent that reason logically A knowledge based agent Representing
More informationLecture 9: Search 8. Victor R. Lesser. CMPSCI 683 Fall 2010
Lecture 9: Search 8 Victor R. Lesser CMPSCI 683 Fall 2010 ANNOUNCEMENTS REMEMBER LECTURE ON TUESDAY! EXAM ON OCTOBER 18 OPEN BOOK ALL MATERIAL COVERED IN LECTURES REQUIRED READINGS WILL MOST PROBABLY NOT
More informationSolvers for the Problem of Boolean Satisfiability (SAT) Will Klieber Aug 31, 2011
Solvers for the Problem of Boolean Satisfiability (SAT) Will Klieber 15-414 Aug 31, 2011 Why study SAT solvers? Many problems reduce to SAT. Formal verification CAD, VLSI Optimization AI, planning, automated
More informationChapter 7 Propositional Satisfiability Techniques
Lecture slides for Automated Planning: Theory and Practice Chapter 7 Propositional Satisfiability Techniques Dana S. Nau CMSC 722, AI Planning University of Maryland, Spring 2008 1 Motivation Propositional
More informationChapter 7 Propositional Satisfiability Techniques
Lecture slides for Automated Planning: Theory and Practice Chapter 7 Propositional Satisfiability Techniques Dana S. Nau University of Maryland 12:58 PM February 15, 2012 1 Motivation Propositional satisfiability:
More informationLogic in AI Chapter 7. Mausam (Based on slides of Dan Weld, Stuart Russell, Subbarao Kambhampati, Dieter Fox, Henry Kautz )
Logic in AI Chapter 7 Mausam (Based on slides of Dan Weld, Stuart Russell, Subbarao Kambhampati, Dieter Fox, Henry Kautz ) 2 Knowledge Representation represent knowledge about the world in a manner that
More informationAn instance of SAT is defined as (X, S)
SAT: Propositional Satisfiability 22c:45 Artificial Intelligence Russell & Norvig, Ch. 7.6 Validity vs. Satisfiability Validity: A sentence is valid if it is true in every interpretation (every interpretation
More informationFoundations of Artificial Intelligence
Foundations of Artificial Intelligence 7. Propositional Logic Rational Thinking, Logic, Resolution Joschka Boedecker and Wolfram Burgard and Bernhard Nebel Albert-Ludwigs-Universität Freiburg May 17, 2016
More informationThe Wumpus Game. Stench Gold. Start. Cao Hoang Tru CSE Faculty - HCMUT
The Wumpus Game Stench Stench Gold Stench Start 1 The Wumpus Game Stench in the square containing the wumpus and in the directly adjacent squares in the squares directly adjacent to a pit Glitter in the
More informationLogic in AI Chapter 7. Mausam (Based on slides of Dan Weld, Stuart Russell, Dieter Fox, Henry Kautz )
Logic in AI Chapter 7 Mausam (Based on slides of Dan Weld, Stuart Russell, Dieter Fox, Henry Kautz ) Knowledge Representation represent knowledge in a manner that facilitates inferencing (i.e. drawing
More informationPropositional Reasoning
Propositional Reasoning CS 440 / ECE 448 Introduction to Artificial Intelligence Instructor: Eyal Amir Grad TAs: Wen Pu, Yonatan Bisk Undergrad TAs: Sam Johnson, Nikhil Johri Spring 2010 Intro to AI (CS
More informationComputational Logic. Davide Martinenghi. Spring Free University of Bozen-Bolzano. Computational Logic Davide Martinenghi (1/30)
Computational Logic Davide Martinenghi Free University of Bozen-Bolzano Spring 2010 Computational Logic Davide Martinenghi (1/30) Propositional Logic - sequent calculus To overcome the problems of natural
More informationPlanning as Satisfiability
Planning as Satisfiability Alan Fern * Review of propositional logic (see chapter 7) Planning as propositional satisfiability Satisfiability techniques (see chapter 7) Combining satisfiability techniques
More informationLecture 2 Propositional Logic & SAT
CS 5110/6110 Rigorous System Design Spring 2017 Jan-17 Lecture 2 Propositional Logic & SAT Zvonimir Rakamarić University of Utah Announcements Homework 1 will be posted soon Propositional logic: Chapter
More informationPropositional Logic: Methods of Proof. Chapter 7, Part II
Propositional Logic: Methods of Proof Chapter 7, Part II Inference in Formal Symbol Systems: Ontology, Representation, ti Inference Formal Symbol Systems Symbols correspond to things/ideas in the world
More informationPrice: $25 (incl. T-Shirt, morning tea and lunch) Visit:
Three days of interesting talks & workshops from industry experts across Australia Explore new computing topics Network with students & employers in Brisbane Price: $25 (incl. T-Shirt, morning tea and
More informationPropositional Logic: Methods of Proof (Part II)
Propositional Logic: Methods of Proof (Part II) This lecture topic: Propositional Logic (two lectures) Chapter 7.1-7.4 (previous lecture, Part I) Chapter 7.5 (this lecture, Part II) (optional: 7.6-7.8)
More informationFoundations of Artificial Intelligence
Foundations of Artificial Intelligence 7. Propositional Logic Rational Thinking, Logic, Resolution Joschka Boedecker and Wolfram Burgard and Frank Hutter and Bernhard Nebel Albert-Ludwigs-Universität Freiburg
More informationLecture 9: The Splitting Method for SAT
Lecture 9: The Splitting Method for SAT 1 Importance of SAT Cook-Levin Theorem: SAT is NP-complete. The reason why SAT is an important problem can be summarized as below: 1. A natural NP-Complete problem.
More informationTopics in Model-Based Reasoning
Towards Integration of Proving and Solving Dipartimento di Informatica Università degli Studi di Verona Verona, Italy March, 2014 Automated reasoning Artificial Intelligence Automated Reasoning Computational
More informationLocal and Stochastic Search
RN, Chapter 4.3 4.4; 7.6 Local and Stochastic Search Some material based on D Lin, B Selman 1 Search Overview Introduction to Search Blind Search Techniques Heuristic Search Techniques Constraint Satisfaction
More informationPropositional and First Order Reasoning
Propositional and First Order Reasoning Terminology Propositional variable: boolean variable (p) Literal: propositional variable or its negation p p Clause: disjunction of literals q \/ p \/ r given by
More informationEE562 ARTIFICIAL INTELLIGENCE FOR ENGINEERS
EE562 ARTIFICIAL INTELLIGENCE FOR ENGINEERS Lecture 10, 5/9/2005 University of Washington, Department of Electrical Engineering Spring 2005 Instructor: Professor Jeff A. Bilmes Logical Agents Chapter 7
More informationArtificial Intelligence
Artificial Intelligence Constraint Satisfaction Vibhav Gogate The University of Texas at Dallas Some material courtesy of Rina Dechter, Alex Ihler and Stuart Russell Constraint Satisfaction Problems The
More informationClause/Term Resolution and Learning in the Evaluation of Quantified Boolean Formulas
Journal of Artificial Intelligence Research 1 (1993) 1-15 Submitted 6/91; published 9/91 Clause/Term Resolution and Learning in the Evaluation of Quantified Boolean Formulas Enrico Giunchiglia Massimo
More informationDeliberative Agents Knowledge Representation I. Deliberative Agents
Deliberative Agents Knowledge Representation I Vasant Honavar Bioinformatics and Computational Biology Program Center for Computational Intelligence, Learning, & Discovery honavar@cs.iastate.edu www.cs.iastate.edu/~honavar/
More informationWorst-Case Upper Bound for (1, 2)-QSAT
Worst-Case Upper Bound for (1, 2)-QSAT Minghao Yin Department of Computer, Northeast Normal University, Changchun, China, 130117 ymh@nenu.edu.cn Abstract. The rigorous theoretical analysis of the algorithm
More informationSAT Solvers: Theory and Practice
Summer School on Verification Technology, Systems & Applications, September 17, 2008 p. 1/98 SAT Solvers: Theory and Practice Clark Barrett barrett@cs.nyu.edu New York University Summer School on Verification
More informationLogical Inference. Artificial Intelligence. Topic 12. Reading: Russell and Norvig, Chapter 7, Section 5
rtificial Intelligence Topic 12 Logical Inference Reading: Russell and Norvig, Chapter 7, Section 5 c Cara MacNish. Includes material c S. Russell & P. Norvig 1995,2003 with permission. CITS4211 Logical
More informationPart 1: Propositional Logic
Part 1: Propositional Logic Literature (also for first-order logic) Schöning: Logik für Informatiker, Spektrum Fitting: First-Order Logic and Automated Theorem Proving, Springer 1 Last time 1.1 Syntax
More informationLogical Agents. Chapter 7
Logical Agents Chapter 7 Outline Knowledge-based agents Wumpus world Logic in general - models and entailment Propositional (Boolean) logic Equivalence, validity, satisfiability Inference rules and theorem
More informationLogical Agents. Santa Clara University
Logical Agents Santa Clara University Logical Agents Humans know things Humans use knowledge to make plans Humans do not act completely reflexive, but reason AI: Simple problem-solving agents have knowledge
More informationImproving Repair-based Constraint Satisfaction Methods by Value Propagation
From: AAAI-94 Proceedings. Copyright 1994, AAAI (www.aaai.org). All rights reserved. Improving Repair-based Constraint Satisfaction Methods by Value Propagation Nobuhiro Yugami Yuiko Ohta Mirotaka Nara
More informationPhase Transitions and Local Search for k-satisfiability
Phase Transitions and Local Search for k-satisfiability Pekka Orponen (joint work with many others) Department of Information and Computer Science Helsinki University of Technology TKK Outline Part I:
More informationWHAT IS AN SMT SOLVER? Jaeheon Yi - April 17, 2008
WHAT IS AN SMT SOLVER? Jaeheon Yi - April 17, 2008 WHAT I LL TALK ABOUT Propositional Logic Terminology, Satisfiability, Decision Procedure First-Order Logic Terminology, Background Theories Satisfiability
More informationFoundations of Artificial Intelligence
Foundations of Artificial Intelligence 7. Propositional Logic Rational Thinking, Logic, Resolution Wolfram Burgard, Maren Bennewitz, and Marco Ragni Albert-Ludwigs-Universität Freiburg Contents 1 Agents
More informationPruning Search Space for Weighted First Order Horn Clause Satisfiability
Pruning Search Space for Weighted First Order Horn Clause Satisfiability Naveen Nair 123, Anandraj Govindan 2, Chander Jayaraman 2, Kiran TVS 2, and Ganesh Ramakrishnan 21 1 IITB-Monash Research Academy,
More informationPrinciples of AI Planning
Principles of 5. Planning as search: progression and regression Malte Helmert and Bernhard Nebel Albert-Ludwigs-Universität Freiburg May 4th, 2010 Planning as (classical) search Introduction Classification
More informationFormal Verification Methods 1: Propositional Logic
Formal Verification Methods 1: Propositional Logic John Harrison Intel Corporation Course overview Propositional logic A resurgence of interest Logic and circuits Normal forms The Davis-Putnam procedure
More informationPropositional Logic: Methods of Proof (Part II)
Propositional Logic: Methods of Proof (Part II) You will be expected to know Basic definitions Inference, derive, sound, complete Conjunctive Normal Form (CNF) Convert a Boolean formula to CNF Do a short
More informationIntroduction. Chapter Context
Chapter 1 Introduction As soon as an Analytical Engine exists, it will necessarily guide the future course of the science. Whenever any result is sought by its aid, the question will then arise But what
More informationPrinciples of AI Planning
Principles of AI Planning 5. Planning as search: progression and regression Albert-Ludwigs-Universität Freiburg Bernhard Nebel and Robert Mattmüller October 30th, 2013 Introduction Classification Planning
More informationCS156: The Calculus of Computation
CS156: The Calculus of Computation Zohar Manna Winter 2010 It is reasonable to hope that the relationship between computation and mathematical logic will be as fruitful in the next century as that between
More informationTitle: Logical Agents AIMA: Chapter 7 (Sections 7.4 and 7.5)
B.Y. Choueiry 1 Instructor s notes #12 Title: Logical Agents AIMA: Chapter 7 (Sections 7.4 and 7.5) Introduction to Artificial Intelligence CSCE 476-876, Fall 2018 URL: www.cse.unl.edu/ choueiry/f18-476-876
More informationCS:4420 Artificial Intelligence
CS:4420 Artificial Intelligence Spring 2018 Propositional Logic Cesare Tinelli The University of Iowa Copyright 2004 18, Cesare Tinelli and Stuart Russell a a These notes were originally developed by Stuart
More informationLogic and Inferences
Artificial Intelligence Logic and Inferences Readings: Chapter 7 of Russell & Norvig. Artificial Intelligence p.1/34 Components of Propositional Logic Logic constants: True (1), and False (0) Propositional
More information7. Propositional Logic. Wolfram Burgard and Bernhard Nebel
Foundations of AI 7. Propositional Logic Rational Thinking, Logic, Resolution Wolfram Burgard and Bernhard Nebel Contents Agents that think rationally The wumpus world Propositional logic: syntax and semantics
More informationCS 188: Artificial Intelligence Spring 2007
CS 188: Artificial Intelligence Spring 2007 Lecture 8: Logical Agents - I 2/8/2007 Srini Narayanan ICSI and UC Berkeley Many slides over the course adapted from Dan Klein, Stuart Russell or Andrew Moore
More informationTutorial 1: Modern SMT Solvers and Verification
University of Illinois at Urbana-Champaign Tutorial 1: Modern SMT Solvers and Verification Sayan Mitra Electrical & Computer Engineering Coordinated Science Laboratory University of Illinois at Urbana
More informationComp487/587 - Boolean Formulas
Comp487/587 - Boolean Formulas 1 Logic and SAT 1.1 What is a Boolean Formula Logic is a way through which we can analyze and reason about simple or complicated events. In particular, we are interested
More informationPropositional Logic. Methods & Tools for Software Engineering (MTSE) Fall Prof. Arie Gurfinkel
Propositional Logic Methods & Tools for Software Engineering (MTSE) Fall 2017 Prof. Arie Gurfinkel References Chpater 1 of Logic for Computer Scientists http://www.springerlink.com/content/978-0-8176-4762-9/
More informationThe Calculus of Computation: Decision Procedures with Applications to Verification. Part I: FOUNDATIONS. by Aaron Bradley Zohar Manna
The Calculus of Computation: Decision Procedures with Applications to Verification Part I: FOUNDATIONS by Aaron Bradley Zohar Manna 1. Propositional Logic(PL) Springer 2007 1-1 1-2 Propositional Logic(PL)
More informationIntelligent Agents. Pınar Yolum Utrecht University
Intelligent Agents Pınar Yolum p.yolum@uu.nl Utrecht University Logical Agents (Based mostly on the course slides from http://aima.cs.berkeley.edu/) Outline Knowledge-based agents Wumpus world Logic in
More informationAlgorithms and Complexity of Constraint Satisfaction Problems (course number 3)
Algorithms and Complexity of Constraint Satisfaction Problems (course number 3) Nicolas (Miki) Hermann LIX, École Polytechnique hermann@lix.polytechnique.fr Miki Hermann Algorithms and Complexity of csp
More informationKnowledge base (KB) = set of sentences in a formal language Declarative approach to building an agent (or other system):
Logic Knowledge-based agents Inference engine Knowledge base Domain-independent algorithms Domain-specific content Knowledge base (KB) = set of sentences in a formal language Declarative approach to building
More informationCS 730/830: Intro AI. 2 handouts: slides, asst 5. Wheeler Ruml (UNH) Lecture 10, CS / 19. What is KR? Prop. Logic. Reasoning
CS 730/830: Intro AI 2 handouts: slides, asst 5 Wheeler Ruml (UNH) Lecture 10, CS 730 1 / 19 History of Logic Advice Taker The PSSH Introduction to Knowledge Representation and Wheeler Ruml (UNH) Lecture
More informationAutomated Program Verification and Testing 15414/15614 Fall 2016 Lecture 3: Practical SAT Solving
Automated Program Verification and Testing 15414/15614 Fall 2016 Lecture 3: Practical SAT Solving Matt Fredrikson mfredrik@cs.cmu.edu October 17, 2016 Matt Fredrikson SAT Solving 1 / 36 Review: Propositional
More informationPropositional logic. Programming and Modal Logic
Propositional logic Programming and Modal Logic 2006-2007 4 Contents Syntax of propositional logic Semantics of propositional logic Semantic entailment Natural deduction proof system Soundness and completeness
More informationHeuristics for Efficient SAT Solving. As implemented in GRASP, Chaff and GSAT.
Heuristics for Efficient SAT Solving As implemented in GRASP, Chaff and GSAT. Formulation of famous problems as SAT: k-coloring (1/2) The K-Coloring problem: Given an undirected graph G(V,E) and a natural
More informationClassical Propositional Logic
Classical Propositional Logic Peter Baumgartner http://users.cecs.anu.edu.au/~baumgart/ Ph: 02 6218 3717 Data61/CSIRO and ANU July 2017 1 / 71 Classical Logic and Reasoning Problems A 1 : Socrates is a
More informationLecture Notes on SAT Solvers & DPLL
15-414: Bug Catching: Automated Program Verification Lecture Notes on SAT Solvers & DPLL Matt Fredrikson André Platzer Carnegie Mellon University Lecture 10 1 Introduction In this lecture we will switch
More informationSearch and Lookahead. Bernhard Nebel, Julien Hué, and Stefan Wölfl. June 4/6, 2012
Search and Lookahead Bernhard Nebel, Julien Hué, and Stefan Wölfl Albert-Ludwigs-Universität Freiburg June 4/6, 2012 Search and Lookahead Enforcing consistency is one way of solving constraint networks:
More informationProof Methods for Propositional Logic
Proof Methods for Propositional Logic Logical equivalence Two sentences are logically equivalent iff they are true in the same models: α ß iff α β and β α Russell and Norvig Chapter 7 CS440 Fall 2015 1
More informationThe Interface between P and NP. Toby Walsh Cork Constraint Computation Centre
The Interface between P and NP Toby Walsh Cork Constraint Computation Centre http://4c.ucc.ie/~tw 3-SAT Where are the hard 3-SAT problems? Sample randomly generated 3-SAT Fix number of clauses, l Number
More informationFrom SAT To SMT: Part 1. Vijay Ganesh MIT
From SAT To SMT: Part 1 Vijay Ganesh MIT Software Engineering & SMT Solvers An Indispensable Tactic for Any Strategy Formal Methods Program Analysis SE Goal: Reliable/Secure Software Automatic Testing
More informationIntroduction to Artificial Intelligence Propositional Logic & SAT Solving. UIUC CS 440 / ECE 448 Professor: Eyal Amir Spring Semester 2010
Introduction to Artificial Intelligence Propositional Logic & SAT Solving UIUC CS 440 / ECE 448 Professor: Eyal Amir Spring Semester 2010 Today Representation in Propositional Logic Semantics & Deduction
More informationSatisfiability Modulo Theories
Satisfiability Modulo Theories Summer School on Formal Methods Menlo College, 2011 Bruno Dutertre and Leonardo de Moura bruno@csl.sri.com, leonardo@microsoft.com SRI International, Microsoft Research SAT/SMT
More informationIntroduction to Artificial Intelligence. Logical Agents
Introduction to Artificial Intelligence Logical Agents (Logic, Deduction, Knowledge Representation) Bernhard Beckert UNIVERSITÄT KOBLENZ-LANDAU Winter Term 2004/2005 B. Beckert: KI für IM p.1 Outline Knowledge-based
More informationClass Assignment Strategies
Class Assignment Strategies ì Team- A'ack: Team a'ack ì Individualis2c: Search for possible ì Poli2cal: look at others and make decision based on who is winning, who is loosing, and conversa;on ì Emo2on
More informationIntroduction Algorithms Applications MINISAT. Niklas Sörensson Chalmers University of Technology and Göteborg University
SAT ALGORITHMS AND APPLICATIONS nik@cschalmersse Chalmers University of Technology and Göteborg University Empirically Successful Classical Automated Reasoning a CADE-20 Workshop 22nd - 23th July, 2005
More informationLogic in AI Chapter 7. Mausam (Based on slides of Dan Weld, Stuart Russell, Subbarao Kambhampati, Dieter Fox, Henry Kautz )
Logic in AI Chapter 7 Mausam (Based on slides of Dan Weld, Stuart Russell, Subbarao Kambhampati, Dieter Fox, Henry Kautz ) 2 Knowledge Representation represent knowledge about the world in a manner that
More informationSolving SAT Modulo Theories
Solving SAT Modulo Theories R. Nieuwenhuis, A. Oliveras, and C.Tinelli. Solving SAT and SAT Modulo Theories: from an Abstract Davis-Putnam-Logemann-Loveland Procedure to DPLL(T) Mooly Sagiv Motivation
More informationExploiting a Theory of Phase Transitions in Three-Satisability. Problems. David M. Pennock and Quentin F. Stout. University ofmichigan
Exploiting a Theory of Phase Transitions in Three-Satisability Problems David M. Pennock and Quentin F. Stout Electrical Engineering and Computer Science Department University ofmichigan Ann Arbor, MI
More informationLOGIC PROPOSITIONAL REASONING
LOGIC PROPOSITIONAL REASONING WS 2017/2018 (342.208) Armin Biere Martina Seidl biere@jku.at martina.seidl@jku.at Institute for Formal Models and Verification Johannes Kepler Universität Linz Version 2018.1
More informationSatisifiability and Probabilistic Satisifiability
Satisifiability and Probabilistic Satisifiability Department of Computer Science Instituto de Matemática e Estatística Universidade de São Paulo 2010 Topics Next Issue The SAT Problem The Centrality of
More informationBackdoor Sets for DLL Subsolvers
Backdoor Sets for DLL Subsolvers Stefan Szeider Department of Computer Science Durham University DH1 3LE Durham, England, UK Abstract We study the parameterized complexity of detecting small backdoor sets
More informationPropositional Logic: Methods of Proof (Part II)
Propositional Logic: Methods of Proof (Part II) You will be expected to know Basic definitions Inference, derive, sound, complete Conjunctive Normal Form (CNF) Convert a Boolean formula to CNF Do a short
More informationThe Constrainedness Knife-Edge. Toby Walsh. APES Group. Glasgow, Scotland.
Abstract A general rule of thumb is to tackle the hardest part of a search problem rst. Many heuristics therefore try to branch on the most constrained variable. To test their eectiveness at this, we measure
More informationCS 380: ARTIFICIAL INTELLIGENCE
CS 380: RTIFICIL INTELLIGENCE PREDICTE LOGICS 11/8/2013 Santiago Ontañón santi@cs.drexel.edu https://www.cs.drexel.edu/~santi/teaching/2013/cs380/intro.html Summary of last day: Logical gents: The can
More informationPropositional Calculus
Propositional Calculus Dr. Neil T. Dantam CSCI-498/598 RPM, Colorado School of Mines Spring 2018 Dantam (Mines CSCI, RPM) Propositional Calculus Spring 2018 1 / 64 Calculus? Definition: Calculus A well
More informationIntroduction to Arti Intelligence
Introduction to Arti Intelligence cial Lecture 4: Constraint satisfaction problems 1 / 48 Constraint satisfaction problems: Today Exploiting the representation of a state to accelerate search. Backtracking.
More informationPropositional and Predicate Logic. jean/gbooks/logic.html
CMSC 630 February 10, 2009 1 Propositional and Predicate Logic Sources J. Gallier. Logic for Computer Science, John Wiley and Sons, Hoboken NJ, 1986. 2003 revised edition available on line at http://www.cis.upenn.edu/
More informationArtificial Intelligence Chapter 7: Logical Agents
Artificial Intelligence Chapter 7: Logical Agents Michael Scherger Department of Computer Science Kent State University February 20, 2006 AI: Chapter 7: Logical Agents 1 Contents Knowledge Based Agents
More informationA Collection of Problems in Propositional Logic
A Collection of Problems in Propositional Logic Hans Kleine Büning SS 2016 Problem 1: SAT (respectively SAT) Instance: A propositional formula α (for SAT in CNF). Question: Is α satisfiable? The problems
More informationCS 380: ARTIFICIAL INTELLIGENCE PREDICATE LOGICS. Santiago Ontañón
CS 380: RTIFICIL INTELLIGENCE PREDICTE LOGICS Santiago Ontañón so367@drexeledu Summary of last day: Logical gents: The can reason from the knowledge they have They can make deductions from their perceptions,
More informationPropositional Logic: Models and Proofs
Propositional Logic: Models and Proofs C. R. Ramakrishnan CSE 505 1 Syntax 2 Model Theory 3 Proof Theory and Resolution Compiled at 11:51 on 2016/11/02 Computing with Logic Propositional Logic CSE 505
More informationMathematical Logic Part Three
Mathematical Logic Part hree riday our Square! oday at 4:15PM, Outside Gates Announcements Problem Set 3 due right now. Problem Set 4 goes out today. Checkpoint due Monday, October 22. Remainder due riday,
More informationNew Inference Rules for Max-SAT
Journal of Artificial Intelligence Research 30 (2007) 321 359 Submitted 11/06; published 10/07 New Inference Rules for Max-SAT Chu Min Li LaRIA, Université de Picardie Jules Verne 33 Rue St. Leu, 80039
More informationArtificial Intelligence
Artificial Intelligence Propositional Logic Marc Toussaint University of Stuttgart Winter 2016/17 (slides based on Stuart Russell s AI course) Motivation: Most students will have learnt about propositional
More informationKnowledge Compilation and Theory Approximation. Henry Kautz and Bart Selman Presented by Kelvin Ku
Knowledge Compilation and Theory Approximation Henry Kautz and Bart Selman Presented by Kelvin Ku kelvin@cs.toronto.edu 1 Overview Problem Approach Horn Theory and Approximation Computing Horn Approximations
More information1 Efficient Transformation to CNF Formulas
1 Efficient Transformation to CNF Formulas We discuss an algorithm, due to Tseitin [?], which efficiently transforms an arbitrary Boolean formula φ to a CNF formula ψ such that ψ has a model if and only
More informationRelaxed Random Search for Solving K-Satisfiability and its Information Theoretic Interpretation
Relaxed Random Search for Solving K-Satisfiability and its Information Theoretic Interpretation Amirahmad Nayyeri Department of Science, Division of Computer Science Salman Farsi University of Kazerun
More informationChapter 6 Constraint Satisfaction Problems
Chapter 6 Constraint Satisfaction Problems CS5811 - Artificial Intelligence Nilufer Onder Department of Computer Science Michigan Technological University Outline CSP problem definition Backtracking search
More informationIntroduction to SAT (constraint) solving. Justyna Petke
Introduction to SAT (constraint) solving Justyna Petke SAT, SMT and CSP solvers are used for solving problems involving constraints. The term constraint solver, however, usually refers to a CSP solver.
More informationThe 30 Years War. Wheeler Ruml (UNH) Lecture 11, CS / 24. Leibniz. Logic in Practice. Satisfiability ILP
The 30 Years War reduction in German population 15 30% in some terrirories 3/4 of the population died male population reduced by almost half population of Czech lands reduced by 1/3 Wheeler Ruml (UNH)
More informationDecision Procedures for Satisfiability and Validity in Propositional Logic
Decision Procedures for Satisfiability and Validity in Propositional Logic Meghdad Ghari Institute for Research in Fundamental Sciences (IPM) School of Mathematics-Isfahan Branch Logic Group http://math.ipm.ac.ir/isfahan/logic-group.htm
More information